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A352288 Total cophenetic index of the rooted tree with Matula-Goebel number n.

%I A352288 #8 Dec 19 2024 11:46:19
%S A352288 0,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,2,0,3,0,1,0,1,0,0,1,0,1,1,0,0,0,0,2,
%T A352288 1,0,3,3,1,0,2,1,4,0,0,1,1,0,2,0,2,1,6,0,0,1,3,1,3,0,3,0,1,0,1,0,6,2,
%U A352288 1,1,3,0,4,3,0,3,1,1,1,0,0,2,2,1,2,4,1
%N A352288 Total cophenetic index of the rooted tree with Matula-Goebel number n.
%C A352288 Mir, Rosselló, and Rotger, define the cophenetic value of a pair of childless vertices as the depth (distance down from the root) of their deepest common ancestor, and they then define the total cophenetic index of a tree as the sum of the cophenetic values over all such pairs.
%C A352288 a(n) = 0 iff n is in A325663, being rooted stars with any arm lengths, since the root (depth 0) is the common ancestor of every childless pair.
%C A352288 An identity relating the childless terminal Wiener index TW(n) = A348959(n) can be constructed by noting it measures distances from a pair of childless vertices to their common ancestor, and the cophenetic values measure from that ancestor up to the root.  So 2*a(n) + TW(n) is total depths Ext(n) = A196048(n) of the childless vertices, repeated by childless vertices C(n) = A109129(n) except itself, so that 2*a(n) + TW(n) = Ext(n)*(C(n) - 1)
%H A352288 Arnau Mir, Francesc Rosselló, and Lucía Rotger, <a href="http://arxiv.org/abs/1202.1223">A New Balance Index for Phylogenetic Trees</a>, arXiv:1202.1223 [q-bio.PE], 2012.
%H A352288 Kevin Ryde, <a href="/A352288/a352288.gp.txt">PARI/GP Code</a>
%H A352288 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A352288 a(n) = Sum_{i=1..k} a(primepi(p[i])) + binomial(C(p[i]),2), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), and C(n) = A109129(n) is the number of childless vertices.
%e A352288 For n=111, the tree and its childless pairs and deepest common ancestors are
%e A352288   root  R         pair  ancestor depth
%e A352288        / \         G,D     A       1
%e A352288       A   B        G,E     A       1
%e A352288      /|\   \       D,E     A       1
%e A352288     C D E   F     any,F    R       0
%e A352288     |                             ---
%e A352288     G                 total a(n) = 3
%o A352288 (PARI) \\ See links.
%Y A352288 Cf. A348959 (terminal Wiener), A196048 (external length), A109129 (childless vertices).
%Y A352288 Cf. A325663 (indices of 0's), A352289 (max by leaves).
%K A352288 nonn
%O A352288 1,17
%A A352288 _Kevin Ryde_, Mar 16 2022